When asking a question you have to use inversion. Also, you agreement is off. Everything needs to agree... so you need either "Lesquelles aiment-elles?" or "Lesquels aiment-ils?" The feminine version of "they" (elles) is only used if the entire group in question is female; if there are 10 billion females and 1 male, it is still "ils." If the gender is unknown, you go with the masculine version; and obviously if it's all male you go with "ils."

So, the reason "Lesquelles ils aiment?" is wrong is twofold: 1. when asking a question you need to use inversion; and 2. The entire sentence needs to agree. Disregarding the improper agreement for a second "Lesquelles [sic] ils aiment?" means "Which ones they love?" By changing it to "Lesquelles aiment-elles?" or "Lesquels aiment-ils?" you get a proper French sentence that means "Which ones do they love?"

I'm not exactly sure about your second point, that they need to agree.
Lesquels/Lesquelles is (in a way) the object, while ils/elles is the subject.
Lesquelles aiment-ils could mean Which (women/other feminine noun) do they (the men) love? and accordingly Lesquels aiment-elles: Which (men/other masculine noun) do they (the women) love. It's totally correct!

"Ils/Elles" = subject
"Lesquels/Lesquelles" = object
Each one has its own gender - one for who loves and the other one for what is loved. Therefore, both constructions you made can be correct, if you're asking whether a group of girls likes watches ("montres", feminine plural) or hats ("chapeaux", masculine plural), for example.

Actually, the inversion for the third-person singular receives a "-t-" if the verb doesn't have one, making it "aime-t-elle" and "aiment-elles" the same in sound - there is no way to distinguish between them.

The hyphen in the question means the subject is coming after an verb, which always happens in English but, in French, is indicated by that sign. That way, "lesquels" is pointing to an object (which ones?), "aiment" is the verb and "elles" is the subject. The object is masculine plural and the subject is feminine plural - that's not a problem as those are different elements in the phrase.